Mol. Cryst. Lip. Crpr , 1997, Vol. 2W.pp. 301-306
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1)ISCLINATION LOOP IN MOM-NAKANJSHI ANSATZ:
ROLE OF THE DIVERGENCE ELASTICITY
O.D. LAVRENTOVICH I , T.ISHIKAWA ' , A N D E.M. TERENTJEV '
Liquid Crystal Institute and Chemical' Physica Program,
Kent State University, Kent, Ohio 44242, USA
'
Cavendish Laboratory, Madingley bad, Cambridge, CB3 OEH, U.K.
Abstract
Energetic stability ofloop defects in nematic liquid crystal ib analyzed
using Mori-Nakaniahi ansats and Rank-0s- theory with non-xrr, Kz4 divergence
term. The ratio of the bulk and divergence &tic
constantrr defines a churackrbtic
material length, which is the equilibrium radius a* of the djscbnation ring. Positive
Ka4 forces the ring to shrink.
Nematic liquid .crystals show a rich variety of defects, including point ('hedgehog"
or umonopolen) and line ("discIinationn) sing&tiesl.
Important peculiarities of
nematic defects are brought about by the identity -n
n; u is the dircctor thai
describes moleculm orient at ion. The identity makes it topologically
for a
monopole to transform into a disclinatiw ring of strength 1/22-7. For a large volume the stability of the hedgehog ve. loop should be d&ed
by intrinsic nematic
parameters such as Frank elastic constants.
Mori and Nakanishi2 conaidered how the hedgehog vs. loop stability depends
on the splay md bend elastic constants. In this article we analyze bow their results
would be modifid by the divergence elastic ttrm
not change
K24.
Although the Ka4 term does
the bulk equiljbrium equations, it has to be taken into acwnnt when
there arc Copologicd changes, such as transformali~nof the ipheical-like core of the
hedgehog into a torus-like core of the disdination ring with a macroscopic radius
much lager than the nematic coherence length ( .
The Frank-Owen free elastic energy densityu
302
O.D.LAVRENTOVICH et 01.
contains the standard contribution with elastic constant K (represented here in tbe
one-constantapproximation') as well as two divergence terms with constnnts KIJand
K2,. We consider only K21 term, assuming K I 3= 0.
The status of the Kzl term as compared to that of the bulk K
terms can be
illustrated with an equilibrium radial hedgehog, n = i in spherical coordinates. To
calc&te
the energy of the hedgehog,one needs to integrate f over r, 5
R > ( is the radius of the system and
r,
-4
iti the
coxe radius
F
< R, w h u e
of the defect; in the
f do- not represent the true energy density. Both K and K24 mntributionn to
the hedgehog's energy are linear in R:
care,
-
-
here Ecmh K/€'is the hedgehog's core energy density with some effective elastic
constant K
K. The r,-terms are negligible as against R-terms. Eq.(2) shows
that K24 should profoundly influence the stability of ddecbe. The mechanism can be
Uustrated by differentid geometry theoremsg. If n ia normal to a family of surfaces
S, then the KzJl distortion is nothing else but
(G = n 1 m 2 , where
twice
and v2 are the two principal
the
Gauss curvature
curvature^)^:
For the radial hedgehog G = l/rz, and one immediately obtains the
Eq.(2). The appearance of a loop would
mean that
G of S
G in
KT4
e n e r g in
the region endored by the
loop is reduced. For example, G = 0 on t be equatorial disc of radiue a. The difference
in elastic energy between the ring and the hedgehog would be
ring to expand if K24< 0, and to shrink if K24 > 0.
.v
aKw, forcing the
For the quantitative consideration, we follow the Mori-Nakanishi ansataZ of
the disclination loop: n = (1,0,0) in ellipsoidal coordinates (&,+,&I connected to
+
+
the Cartesian ones by x 2 = a Z ( l u2)(1- 7 ' ) cosZ d; p2 = aZ(l u2)(l- r2)sin2 4;
a = aor.
The ansatz asymptotically satisfies boundary conditions n = i and the
eqdibrium condition VZn = 0 as R
which satisfies
exactly the two
+
ao. The limit a = 0 is the radial hedgehog
conditions above. With a
a circular wedge loop of radius a
#
0, the ansatz describes
with n perpendicular to the surface of the oblate
ellipsoids of revolution (Fig.1). Since with a # 0 the ansatz satisfies the equilibrium
ROLE OF THE DIVERGENCE ELASTICITY
303
condition only asymptotically, the wrrmponding energy Fr, teptcsents the upper
limit of minimum energy of the loop2.
Fig.1 Loop of disclination 1/2
To find the equilibrium radius a' of the loop, one has to find the integral Flw
of f and then minimize it. Tht volume of integration should not include the tomlike care of the loop with metidiond radius +, where deformations art large. The
ududGdmlmuccanbec6o~cnu~
0 < o S p , O 5 Irl 5 q,vhcrcp=
-
and p s
en-
d-.
by
- %K,
-d
-
Any other geometry of the mrc crass-section chimges the
which H negligible in comparison with terms
a$ and
Ilk, when
the l i d t of validity oi our consideration.
We fimt obtain the energy F2qcaused by the divewnce term:
a 3 r,; the last i n e q d t y sets
As it w m urpaetcd,the mre p, 9-terrae bring only small wmctions T , K ~The
~ , leading
terms (-8xKMR) and 2aZaKz4art arc-profile-independent and do not vaniah even
when r,/a
.
4
0, th& origin being purely topologicd, as discussed above.
this topdogicd argument hold8 only for a
>f
Of coprsc,
when the region inside the ring is
composed of uniaxial nematic phase.
The total energy FL, of the Imp includes also the K-term and the core energy
2raECJwhere &$, = EcE,s<'/K.In the limit rc/a + 0,
O.D. LAVRENTOVICH e t d .
304
-
Within the framework of Frank- Oseen theory, r, should be estimated 86 r, (
and its scaled energy as E,r
1'. Tha same order of GJ follows from the models
-
of hiaxial core, where vC is larger but ECCis
srndleP*'O.".
Note that the usual b&
terms produce divergent factor# such as Jl ln (a]~,). In contraat, Kz4 integral# bring
--
no divergence, Eq.(4); the same behavior waa found for mnectic focal conic defect~'~.
brings rather strong exponential dependence of the loop radius on
Thia
the ratio of elastic wnstantr. Minimixation of
gives
T h i ~value of a* would be modified by the iadusion of different vatues of splay
Kll and bend Km elagtic condant s. Tn the first approximation of small misotoropy
a = ( K l 1 - K33)/(K,I
+ K:v,:l)the factor is exp(1 f a),and thus K33> KI1favors a
larger loop.
Aa recent experiment^'^-'-'^ and molecular theories16 suggest, K2,4c m be of the
order of
K . According to the Ericksen's stability
criteria, Kz4 6hould be positive'7.
Therefore ~e appenxauce of the loop insttad of the rodid hedgehog inmema the
elastic energy. We conclude that with positive K2, ,the macro~wpicloop is t d h l y
to be
stable as against a radial hedgehog: in &.(6), the radium of the Imp is the order
of few tens of 4 when Kal
course, for as
-
= 0 , but quickly decreaacs to a*
-E
when K*4 -+ K
+
Of
4 the Rank-Oseen approach gives only indicative resdts; a dgwoas
treatment shuuld include t hc gradients of the d e p of otder and psibility of biaxial
regions at the defect core.
Thc Bitnation can be different for hyperbolic htdgehcgs, n = ( - z l r , - y / r , t / r ) ,
r=
y2
+ z 2 . It is easy to see that the elastic energy of the byperbdc hedgehog
containa Ka4 term with a aign opposite to
The
difference in the ~ i g nof the
that of radial hedgehogs:
Kz 1 term
ia b a u a e of predominantly
negative
Gsuseian cumatace of surfacer normal to n ; for the radial hedgehog these ~ n r h e a
are spheres with a podtiye curvature. l'ransfonnaiion of the by perbotic hedgehog into
a loop of m = -1/2 and
radius a should decrease the enera by a quantity
Therefore, the increase of a positive constant K24 would in-
.Y
aK=.
the radius oi the
'
.
-
loop, ra*
ROLE OF THE DIVERGENCE ELASTICITY
p(cxp(AKz4/K); here p
-
305
10 and A -- 1 are the two positive numerical
cotffi&nts with values that can be determined within the Rank-Oaeen approach.
In nnmary, using the Mori-Nakanishi snsatr, we have shown that the saddlesplay elastic term in Frank-Oseen theory with a positive elastic constant KZ4should
force a loop m = 112 and large macroscopic radius (shown in F
i
g
.1) to shrink into a
radial-like structure with a micro~copiccore.
We benefit horn the discussion with V.M. Pergamenshchilr, T.J. Sluckin, J. Toner,
H. Toyoki and 3.R. Trebia. O.D.L. thanks NSF ALSOM Center for support under
pant DMR-20147as well as the organizers of the program 'Topological Defectsn at
Newton Institute far Mathematical Science for hospitality. E.M.T's research
the 1
-
h u been support4 by Unilever PLC.
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